Question

Let $T_n$ denote the number of triangles which can be formed by using the vertices of a regular polygon of $n$ sides. If $T_{n+1}-T_n=28,$ then $n$ equals

• A. 4
• B. 5
• C. 6
• D. 7
• E. 8

Answer 1

Answer: (E)

Given, $T_{n+1}-T_n=28$
$\Rightarrow$ ${}^{n+1}{C}_3{-}^n{C}_3=28$
$\Rightarrow$ $\frac{(n+1)!}{3!(n-2)!}-\frac{n!}{(n-3)!3!}=28$
$\Rightarrow$ $\frac{1}{6}[(n+1)(n)(n-1)-n(n-1)(n-2)]=28$
$\Rightarrow$ $n[n^2-1-(n^2-3n+2)]=168$
$\Rightarrow$ $n^2-n-56=0$
$\Rightarrow$ $n=8$ $(n\ne -7)$